Enigma 1766: Triangular sums
11 September 2013
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From New Scientist #2934, 14th September 2013 [link]
I have listed in random order five positive integers, four of two digits and one of one digit. They use each of the digits 1 to 9. None of the two-digit integers has any factor greater than 1 in common with any other of the two-digit integers. The first of the integers in my list is a triangular number. The sum of the first two, the sum of the first three, the sum of the first four and the sum of all five are also triangular numbers.
What are my integers in the order in which I have listed them?